How Sridhar Thinks

Suppose we are intereted in constructing a magma with the following properties: There is an identity element 1, as well as another (distinct) element -1. -1 squares to 1, and its left and right multiplication action commute with every other left or right multiplication action. Ca…

Consider the fruit puzzle, where we try to solve a/(b + c) + b/(a + c) + c/(a + b) = 4. After moving everything to one side and clearing the denominators, this becomes f(a, b, c) = 0 where f(a, b, c) = a^3 + b^3 + c^3 - 3(a^2 b + a b^2 + a^2 c + a c^2 + b^2 c + b c^2) - 5 a b c. …

I'm surrounded by people talking about this all the time. I feel it turns on what one considers "infallible predictor" to amount to. At any rate, the problem as given above poses that the predictor is infallible, but we can quibble with what that amounts to and what sense it make…

[TODO: Markup] Recall that there is a one-dimensional space of even homogenous distributions of degree -s, for each complex s. (Similarly for odd homogenous distributions, but we will not be concerned with these). When s is not a positive odd integer, these are the constants time…

Assorted results about Golay codes: Given two points in 2^N (i.e., two functions from an N-element set to the 2-element field), their Hamming distance is the number of coordinates (i.e., inputs) on which they differ. By weight, I mean Hamming distance from the zero vector. By th…

What I'd like to say about proofs of Fermat's little theorem (or the Euler-Fermat theorem more generally) is going to be annoying to write on Twitter with its character count restrictions, lack of editing, etc, but here is the gist of it: I've been interested for a long time in c…

Here are two framings and proofs of Zorn's Lemma: (Throughout the following, given a set or predicate S and an element x, we write S_{< x} to mean those elements of or satisfying S which are < x.) In any poset whose underlying set is well-orderable, there is a chain with n…

Various random facts about comma categories I wished to record somewhere: The Comma-Kan lemma from my thesis: TODO The Comma-Kan lemma decomposes as the result of some fact about products and some fact about inserters (from which comma objects can be constructed), at least in a …

Calculus is about certain kinds of weighted sums. An easy kind of weighted sum against a function f is to extract the value f(x) for a specific x. This is like integrating against a Dirac delta. We might also consider the integral of f along some interval (or across some region m…

The mean number of points in cycles of length 𝑘 over all permutations of an 𝑛-element set is 1. Why is this? Well, a point in a cycle of length k is specified by first choosing k many elements to be in the cycle (in one of (n choose k) many ways), then putting an ordering on thes…

The Gram-Euler theorem says that if you have a contractible polytope with straight sides, the sum of the interior solid angles at each cell (where this means the fraction of an infinitesimal sphere at that cell which is contained in the inside of the polytope), counted positively…

Some facts about Gaussian functions (exponentiated quadratics): Let g(x) = Exp(x^2) for some base of exponentiation. Note that g(ax) = g(x)^a^2. In other words, log(g(sqrt(x))) is a linear function of x. This is characteristic of g, in that any f such that f(ax) = f(x)^a^2 is suc…

UNPOLISHED: Some proofs of some facts about normalization in the typed lambda calculus (terms in the simply typed lambda calculus with products, possibly with NNO as well, are normalizing, or possibly strongly normalizing, with unique normal form): A) We can use confluence with "…

UNPOLISHED: Eventually I will record all my thoughts on the Fourier transform here. For now, I just want to record some thoughts about the Fourier inversion theorem. The Fourier transform is characterized by the fact that the Dirac delta as input is turned into the constantly 1 f…

If a monad has a left adjoint, then this left adjoint is a comonad with the same Kleisli category and same inclusion of the pure category into the Kleisli category. If a monad has a right adjoint, then this right adjoint is a comonad with the same Eilenberg-Moore category and the…

Supply chain phenomena to write about: – Economic order quantity – Newsvendor (The above two don’t seem to relate to general phenomena but should still be written. EOQ sort of sets up (s, S) policies, I guess. Newsvendor can be immediately solved by the general theory of convex o…

Salvage the following posts from SDMB: https://boards.straightdope.com/t/macro-quantum-effect-witnessed-implies-time-travel-multiple-universes/535583/41?u=indistinguishable https://boards.straightdope.com/t/entanglement-question-how-to-prove-that-before-one-of-a-pair-is-measured-…

Consider the free theory with an internal X extending x. When we externalize its internal X, we get some X extending x. But is it x on the dot, or might it be changed from x to have more in it? It seems obvious but not obviously obvious that, when Set itself is an X, we should ge…

TODO: Scratch notes for a post on Lawvere theories, monads, functorial semantics, categorical logic, etc. Taken from a Twitter DM conversation. A Lawvere theory, in the sense I like to think of it though others take different views, is just a category with products. What kind of …

The order $N$ associahedron is an abstract polytope whose vertices are all binary trees with $N$ many leaf nodes (where a binary tree is either a leaf node or an ordered pair of two child binary trees). One can describe this same thing in many different words, as having to do wit…

There are lots of things to say about quadratic forms (or equivalently when division by 2 is available, symmetric bilinear forms, or the symmetric component of arbitrary bilinear forms), a notion which comes up over and over in math. Particularly positive-definite ones (which are…

Here are a couple little curiosities in calculus (known since at least 1697 by Johan Bernoulli). They are sometimes called "sophomore's dream", because they feel too good to be true, like the kind of glib substitution of one thing by another somewhat similar thing that a naive st…

I don't really care for Clifford algebra, but here's my attempt to understand the parts of it that aren't obvious anyway: TODO: Write out why I don't care for them. What's the value of bundling together dot product and cross product as even and odd components of a shared product?…

Extracted from Facebook comments: Chenyu Zhao Fun problem of the day: develop an algorithm that samples a Brownian Bridge (a Wiener process s.t. W(0) =W(1) = 0, i.e. the ends are fixed). Richard Starfield Simulate regular Wiener process. Subtract linear function to fix ends. Prof…

TODO: To be written out into a full thing on how Boolean algebra is the finitary theory of finite sets, of 2, etc. Limit vs. product theory perspective on Boolean algebra, etc. But for now, I wanted to record this observation about Boolean rings: Why does adding the condition x^2…

Let $$B$$ be a value such that $$B^N = 1$$ and define the $$K$$-weighted base-$$B$$ transform of order $$N$$ of a function $$f$$ to be $$s \mapsto K \sum_{t = 0}^{N - 1} f(t) B^{st}$$. [The domains of these functions should be thought of as the integers modulo $$N$$, so that thes…

Suppose you play a game: You have a bidirectionally infinite sequence of coins, heads or tails. Your goal is to make this into some particular target sequence. At any moment, you announce some set of positions that you wish to flip. Your opponent, however, can respond by translat…

[TODO: Equivalence between hom-set definition of adjunctions (including as abstracted to any 2-category) and unit/counit/triangle identities definition. Automatic equivalence also to the “backwards” definition: An adjunction between functors F : C → D (the left adjoint) and G : D…

Key concepts: Preorders, (higher-)groupoids. As their intersection, setoids. As their union, (higher-)categories. A preorder: You are given various points with names, and various edges between pairs of points. Edges come with an orientation; a left endpoint and a right endpoint. …

[There are a million things to write about adjunctions. For now, I’m just writing some motley scratch notes for myself and will shape this up into an introductory post later.] Much like every monad can be seen as coming from an adjunction in multiple ways, with Kleisli and Eilenb…

This is all TODO notes for a post not yet actually written, but scratch thoughts I want to keep archived for myself to write with. Let’s start with just the SP theorem, naively without size considerations. In categorical terms, this essentially states that if a category has arbit…